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Normal are drawn to the hyperbola (x^2)/...

Normal are drawn to the hyperbola `(x^2)/(a^2)-(y^2)/(b^2)=1` at point `theta_1` and `theta_2` meeting the conjugate axis at `G_1a n dG_2,` respectively. If `theta_1+theta_2=pi/2,` prove that `C G_1*C G_2=(a^2e^4)/(e^2-1)` , where `C` is the center of the hyperbola and `e` is the eccentricity.

Text Solution

Verified by Experts

The normal at point `P(a sec theta_(1), b tan theta_(2))` is
`ax cos theta_(1)+by cot theta_(1)=(a^(2)+b^(2))`
It meets the conjugate axis at `G_(1)(0,(a^(2)+b^(2))/(b)tan theta_(1))`.
The normal at point `Q(a sec theta_(2), b tan theta_(2))` is
`axcos theta_(2)+" by " cot theta_(2)=(a^(2)+b^(2))`
It meets the conjugate axis at `G_(1)(0,(a^(2)+b^(2))/(b)tan theta_(2)).` Therefore,
`CG_(1)*CG_(2)=((a^(2+b^(2)))^(2))/(b^(2))tantheta_(1)tantheta_(2)`
`=((a^(2)+b^(2)))/(b^(2))" "(becausetheta_(1)+theta_(2)=(pi)/(2))`
`=(a^(4)(1+(b^(2))/(a^(2))))/(b^(2))`
`=(a^(2)e^(4))/(e^(2)-1)`
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